According to Calvinism, what should one do if one doubts their election? What does the reformed church teach in terms of what to do, or pray for, if one begins to suspect that they are not predestined for salvation. 

And if they are in fact not elected by God, does the church have a process for them to discover this about themselves, or are they to continue in hope even though there is no chance of salvation?
Just because a person is a long time church member does not mean they belong to Christ Jesus or that they have put their faith and trust in Him.  There are many people who say they are Christians but who have never experienced the new birth.  Forgetting about Calvin for one moment, born-again Christians look to Christ Jesus and believe in this promise in John 6:37-40:

All that the Father gives me will come to me, and whoever comes to me I will never drive away... I shall lose none of all that he has given me... everyone who looks to the Son and believes in him shall have eternal life, and I will raise him up at the last day.

No if’s, no but’s and no maybe’s.  If a person has been born-again then they have the indwelling of the Holy Spirit and that is their assurance.  If they have the indwelling of the Holy Spirit, then they begin to produce the fruit of the spirit.  A transformed life is evidence of the new birth.   As for seekers who have not yet come to saving faith in Christ Jesus, and who look for assurance, then this extract from a sermon preached by Baptist minister Charles H. Spurgeon on the subject of Election may be helpful:

Many persons want to know their election before they look to Christ, but that is not possible; it is only to be discovered by &quot;looking to Jesus&quot; (Hebrews 12:2). Do you feel yourself to be a lost, guilty sinner? Go straight to the cross of Christ, and tell Jesus so, and tell Him that you have read in the Bible, &quot;Whoever comes to me I will never cast out&quot; (John 6:37). Tell Him that He has said, &quot;The saying is trustworthy and deserving of full acceptance, that Christ Jesus came into the world to save sinners&quot; (1 Timothy 1:15). Look to Jesus and believe on Him, and you shall make proof of your election directly, for as surely as you believe, you are elect.
If you will give yourself wholly up to Christ and trust Him, then you are one of God's chosen ones; but if you stop and say, &quot;I want to know first whether I am elect&quot;, you do not know what you are asking. Go to Jesus, just as you are, in all your guilt. Leave all curious inquiry about election alone. Go straight to Christ, and hide in His wounds, and you shall know your election. The assurance of the Holy Spirit shall be given to you, so that you shall be able to say, &quot;I know whom I have believed, and I am convinced that he is able to guard until that Day what has been entrusted to me&quot; (2 Timothy 1:12).
Christ was at the everlasting council - He can tell you whether you were chosen or not; but you cannot find it out in any other way. Go and put your trust in Him, and His answer will be, &quot;I have loved you with an everlasting love; therefore I have continued my faithfulness to you&quot; (Jeremiah 31:3). There will be no doubt about His having chosen you when you have chosen Him.  (Source: Charles H. Spurgeon’s devotional Morning and Evening, morning 17 July)

This, I believe, expresses the Reformed Protestant view.  As the article says, if a person doesn’t know whether they are part of God’s elect then they need to go to the cross and turn to Christ Jesus, the author and finisher of our faith (Hebrews 12:2-3).  Assurance is experienced by the indwelling of the Holy Spirit, by whom the believer realises that they have been adopted into God’s family:

Romans 8:9, 11, 14:16: “And if anyone does not have the Spirit of Christ, he does not belong to Christ...  And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies through his Spirit, who lives in you...  because those who are led by the Spirit of God are sons of God... And by him we cry, ‘Abba, Father.’  The Spirit himself testifies with our spirit that we are God’s children. “
1 Thessalonians 1:4-6: “For we know, brothers loved by God, that he has chosen you, because our gospel came to you not simply with words, but also with power, with the Holy Spirit and with deep conviction... you welcomed the message with the joy given by the Holy Spirit.”
2 Timothy 1:12: “I know whom I have believed, and am convinced that he is able to guard what I have entrusted to him for that day.”

It is possible for a believer to back-slide, to grieve the Holy Spirit and to become spiritually sick, but that does not mean they are no longer part of God’s elect or that they have lost their salvation.  It means they need to repent and turn to the great physician and cry out for help:

2 Peter 1:1-10: His divine power has given us everything we need for life and godliness through our knowledge of him who called us by his own glory and goodness...  But if anyone does not have them, he is short-sighted and blind, and has forgotten that he has been cleansed from his past sins. Therefore, my brothers, be all the more eager to make your calling and election sure.”

I confess I find all this talk about Calvin and the Elect confusing.  I am a Christian of the Reformed Protestant persuasion but I do not follow Calvin.  I follow Christ and I believe what the Bible tells us.  Here is a brief summary regarding the biblical basis for the believer’s assurance of salvation:

I have been saved – in the past – from the penalty of sin – by a crucified Saviour: “For in this hope we were saved&quot; (Romans 8:24). “For it is by grace you have been saved, through faith – and this not from yourselves, it is the gift of God” (Romans 2:8).
I am being saved – in the present – from the power of sin – by a living Saviour: &quot;For the message of the cross is foolishness to those who are perishing, but to us who are being saved it is the power of God&quot; (1 Corinthians 1:18).
I shall be saved – in the future – from the presence of sin – by a coming Saviour: &quot;Since we have now been justified by his blood, how much more shall we be saved from God's wrath through him!&quot;  (Romans 5:9)

Regeneration is an on-going process but the Christian who belongs to Christ Jesus can be assured that Jesus shall lose none of all that the Father has given him: “everyone who looks to the Son and believes in him shall have eternal life, and I will raise him up at the last day” (John 6:37-40).